User blog:Rgetar/Classes of ordinals ｛-｝a, ｛*｝a, ｛+｝a
Recently I was trying to extend set X{·}a (see Definitions update blog) beyond ΩΩΩΩ.... (Now I like to represent arrays of ordinals as "larger" ordinals). I did not want to give up X{·}a, since it was used in short and independent of fundamental sequence systems definitions of ordinal array functions Xa and generalized Veblen function φ(X). Also, fundamental sequences was not used in these definitions. When I failed to extend X{·}a, I started to formulate equal values and comparison of generalized Veblen function in terms of "larger" ordinals instead of arrays of ordinals. I noticed that apparently it does work beyond ΩΩΩΩ.... Then I suddenly realized, that sets used there can be used instead of X{·}a. (Later I found out that I already used something like this in one of my earliest blogs, but I almost forgot it). {-}βa {-}βa, where a < Ωβ, is class of ordinals. It depends on ordinal a and ordinal β. And it is intersection of classes {-n}βa for all n. {-n}βa are also classes of ordinals such as {-1}βa is subclass of {-0}βa, {-2}βa is subclass of {-1}βa, {-3}βa is subclass of {-2}βa etc. But currently I do not define {-n}βa for all n. Some {-n}βa for large n may be defined later. But for not very lagre ordinals we can use {-n}βa only for small n. Any ordinal can be represented as sum of Ωβici terms, ci < Ωβ. (Special case for β = 0 is Cantor normal form). {-0}βa is class of ordinals such as any ci < a {-1}βa is class of ordinals such as any ci < a, and i ∈ {-1}βa For an ordinal less than Ωβ + 1 term Ωβici can be represented as φβγ(X)c where φβγ(X) is Veblen-like function. For an ordinal of cardinality Ωδ > Ωβ term Ωδici can be represented as φδγ(X)c then we can represent coefficient c the same way and so on until we get all coefficients less than Ωβ. {-2}βa is class of ordinals such as any c < a, and any X ∈ {-2}βa {-3}βa is class of ordinals such as any c < a, and any X and γ ∈ {-3}βa Currently I did not define {-n}βa for n > 3. {*}βa {*}βa is intersection of {*n}βa for all n. {*0}βa is class of ordinals such as it is sum of an ordinal of class {-0}βa and Ωβba for any b {*1}βa is class of ordinals such as it is sum of an ordinal of class {-1}βa and ((Ωβba for b of class {-1}βa) or (Ωβb for b of class {*1}βa)) {*2}βa is class of ordinals such as it is sum of an ordinal of class {-2}βa and ((φδγ(X)c for c of class {*2}βa and X of class {-2}βa) or (φδγ(X) for X of class {*2}βa)) {*3}βa is class of ordinals such as it is sum of an ordinal of class {-3}βa and ((φδγ(X)c for c of class {*2}βa and X and γ of class {-2}βa) or (φδγ(X) for X of class {*2}βa and γ of class {-2}βa) or (φδγ(0) for γ of class {*2}βa)) Currently I did not define {*n}βa for n > 3. {+}βa {+}βa is intersection of {+n}βa for all n. {+n}βa is union of {-n}βa and {*n}βa. X{-}βa X{-}βa is set of ordinals of class {-}βa less than X. X{*}βa X{*}βa is set of ordinals of class {*}βa less than X. X{+}βa X{+}βa is set of ordinals of class {+}βa less than X. And this set can replace X{·}a. Ordinal array function 0βa = a + 1 Xβa = sup(-1; X0)βYβa), Y ∈ X{+}βa Expanded form: 0βa = a + 1 Xβa = sup(X0βYβa), X is successor ordinal, Y ∈ X{+}βa, where X0 = lest(X; 0) Xβa = sup(Yβa), X is limit ordinal, Y ∈ X{+}βa Veblen-like function φαβ(X) = φαγ(φγβ(X)) φαα + 1(X) = ΩαX, if card(X) < Ωα + 1 φαα + 1(X) = α is (1 + leo(X))-th common fixed point of all functions α = φαα + 1(Y), Y ∈ X{+}αa, if card(X) = Ωα + 1 Note: I am not sure, if this enough for limit β. But I guess yes. Note: Veblen-like function is defined using X{+}αa, and X{+}αa is defined using Veblen-like function. But I think this does not cause problem, since for some Veblen-like functions it is enough to use class X{+1}αa, which is defined without using Veblen-like functions. Equal values φαβ(X) = φαβ(Y) where Y ∈ X{*}αφαβ(X) Comparison If Y ∈ X{-}αφαβ(X) then φαβ(X) > φαβ(Y) Standard form φαβ(X) is in standard form, if X is of class {-}αφαβ(X). Otherwise X is of class {*}αφαβ(X), and φαβ(X) is in non-standard form. Fundamental sequences In previous blog I defined fundamental sequence system (fss)-dependent ordinal array function 0βα = α + 1 + 1βα = X0βXβα Xβα = sup([Xn]βα), 1 < cof(X) < Ωβ Xβα = [Xα]βα, cof(X) = Ωβ where α < Ωβ, X < Ωβ + 1, Xβα < Ωβ Here I defined fss-independent ordinal array function. Maybe, this can help to define certain fss (or class of fss's) such as both ordinal array function coincide. Category:Blog posts